How Random Is the M\"obius Function? Smoothing, Probability, and the Riemann Hypothesis

Abstract

This article is primarily expository, but it also contains several new observations and reformulations concerning the M\"obius function, probabilistic models, dynamical systems, and the Riemann hypothesis. Its starting question is classical: in what sense can the M\"obius function be said to behave randomly? We begin with Denjoy's random-walk heuristic and place it in the context of later work on random multiplicative functions, short intervals, and M\"obius pseudorandomness. We then develop two smoothing forms of the classical Mertens criterion for the Riemann hypothesis. The first uses the discrete Laplace transform \[ \Phi(t)=\sum_{n\geq1}\mu(n)e^{-nt} \] and identifies RH with the condition \[ \Phi\in L^p(0,\infty) \qquad\text{for every }1\leq p<2. \] The second uses normalized M\"obius Fourier polynomials and local moments on arcs of length comparable with \(1/N\). The paper also revisits the author's earlier criterion for RH in terms of discrete measures, as presented in Broughan's account of analytic equivalents of RH. Denjoy's heuristic is made precise in a simple independent coefficient model, and this model is carefully distinguished from the modern theory of random multiplicative functions. The resulting coefficient space also gives a measure--category contrast: the relevant \(L^p\)-property has full measure but is topologically meagre. A further point of the paper is dynamical: the multiplicative semigroup of positive integers acts naturally on the coefficient space, and the M\"obius sequence is a distinguished arithmetic point for this action. The aim throughout is explanatory, while keeping these new observations visible: to show how arithmetic, probability, Fourier analysis, Mellin transforms, and dynamics fit together, and to indicate what modern results add to the older random-walk picture.